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3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials...

3. (50) Let f(x) = x^4 + 2. Find a factorization of f(x) into irreducible polynomials in each of the following rings, justifying your answers briefly:

(i) Z3 [x];

(ii) Q[x] (this can be done easily using an appropriate theorem);

(iii) R[x] (hints: you may find it helpful to write γ = 2^(1/4), the positive real fourth root of 2, and to consider factors of the form x^2 + a*x + 2^(1/2);

(iv) C[x] (you may leave your answer in exponential form).

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